Model for Predicting Horizontal Well Transient Productivity in the Bottom-Water Reservoir with Finite Water Bodies

Abstract

To better understand the horizontal well transient productivity in the bottom-water reservoir with finite water bodies, the horizontal well transient productivity model for the bottom-water reservoir with finite water-body multiple was developed using Green’s function and potential superposition method. Laplace transforms, Fourier transforms, superposition of point source, and Duhamel principle were used to obtain the transient productivity of the horizontal well, and the transient productivity of the horizontal well in real space was obtained by the Stehfest numerical inversion method. The typical pressure response curve and dimensionless productivity curves were plotted. The effects of the water-body multiple, the distance between the horizontal well and oil–water contact, and the skin factor, were analyzed. Six main flowing stages were divided for horizontal wells in the bottom-water reservoir with finite water bodies. When the water body multiples are zero or tend to infinity, the results obtained from the model are consistent with the calculations by the conventional top-bottom closed reservoir model or infinite rigid bottom-water reservoir model, respectively, and the pressure dynamic for the finite water body falls in between both. With the increase in the water body multiples and the decrease in distance between the horizontal well and the oil–water contact, and the horizontal well productivity decreases slowly. With the increase in the skin factor, the initial productivity decreases; moreover, the skin factor has a great influence on the initial productivity of the horizontal well, while the later influence gradually decreases. Accurate horizontal well productivity prediction in the bottom-water reservoir with finite water bodies provides a strong basis for horizontal well deployment, design optimization, and formulation of development policy.

1. Introduction

At present, scholars at home and abroad have conducted a large number of studies on horizontal well production prediction in bottom-water reservoirs, mainly for infinite rigid bottom-water reservoirs [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. However, there are fewer studies on horizontal well production in bottom-water reservoirs with finite water bodies.

Studies of steady-state production from horizontal wells in bottom-water reservoirs date back to the 1980s, and current research is more mature than before. Foreign scholars such as Giger and Kuchuk studied earlier in this field [1,2]. Giger proposed a critical production formula for bottom-water reservoirs in 1986 [1]. Kuchuk et al. derived a production formula for a horizontal well in an infinitely extended bottom-water reservoir in the horizontal direction in 1988 by assuming a constant pressure boundary and a closed boundary in the reservoir and using the cosine transformation method [2]. Following this, many researchers considered the impact of factors such as seepage barriers on the critical production of horizontal wells in bottom-water reservoirs on this basis [3,4,5]. Domestic scholar Fan Zifei studied earlier [6,7]. Fan Zifei derived the production equation for a horizontal well in an infinitely extended bottom water-driven reservoir in the horizontal direction in 1993, taking into account the influence of the closed boundary at the top of the reservoir, the constant pressure boundary, the anisotropy of the reservoir, and the horizontal well location on the horizontal well potential distribution in the derivation process. After this, academics such as Cheng Songlin, Dou Hongen, and Chen Yuanqian studied the critical production model for horizontal wells in bottom-water reservoirs using different methods and from different perspectives [8,9,10,11,12]. The steady-state production theory for horizontal wells in bottom-water reservoirs is mainly for infinite rigid water reservoirs and does not apply to the initial production prediction of horizontal wells in bottom-water reservoirs with finite water bodies, nor can it predict the production variation pattern.

Studies of transient productivity from horizontal wells in bottom-water reservoirs date back to the late 1980s. For foreign research, Dikken was the first to use the wellbore friction coefficient formula in conventional pipe flow to calculate horizontal wellbore friction losses in 1989 [13], assuming a constant fluid production index per unit wellbore length in horizontal wells, linking reservoir seepage to wellbore flow through the mass conservation equation in the wellbore, and establishing a transient productivity model. After this, many scholars introduced acceleration pressure drop and proposed a wellbore pressure drop calculation model. They comprehensively considered the flow characteristics in the horizontal wellbore and regarded the friction coefficient as a variable to establish a horizontal well segmental coupling model [14,15,16,17,18,19,20,21]. Domestic scholars have also studied the transient productivity of horizontal wells in bottom-water reservoirs in greater depth. The research methods include mainly analysis methods, physical simulations, and numerical simulations [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. Liu Xiangping, Xiong Jun, Zheng Qiang, Zhou Daiyu, and other researchers established a coupled model of seepage flow and wellbore pressure drop in unsteady reservoirs and carried out analytical calculations of transient productivity in horizontal wells in bottom-water reservoirs and analyzed the variation patterns of indicators such as production along the wellbore and wellhead. Green’s function, source function, Laplace transform, and orthogonal transform were used, and conditions including a segmented bare hole in horizontal wells, shot hole completion, variable density shot hole, and reservoir inhomogeneity were considered [22,23,24,25,26,27,28]. Wang Jialu and Liu Xinying studied the variation law of production and bottom-water ridge entry of horizontal wells in bottom-water reservoirs through indoor physical simulation experiments [29,30,31]. Cui Chuanzhi, Jiang Hanqiao, and other scholars studied the characteristic laws of horizontal well production and water flooding in bottom-water reservoirs using reservoir numerical simulation [32,33,34,35,36,37]. These studies have all been conducted on infinite rigid water reservoirs. They have not considered the effect of finite water-body multiples on horizontal well production and do not apply to horizontal well production prediction in the bottom-water reservoir with finite water bodies.

This paper establishes an unstable production model for horizontal wells in bottom-water reservoirs with finite water bodies with respect to the characteristics of bottom-water reservoirs with finite water bodies, makes dimensionless productivity curves and analyzes the factors influencing production, which can provide a decision basis for the horizontal well deployment, design optimization, and development technology policy formulation in bottom-water reservoirs with finite water bodies.

2. Model Building

2.1. Physical Model

A schematic diagram of the horizontal well development model for a bottom-water reservoir with finite water bodies is shown in Figure 1. The reservoir is laterally infinite. Additionally, the reservoir is vertically divided into the following two parts: the upper part is the unutilized oil layer, and the lower part is the water layer. The upper and lower layers have different pore permeability characteristics.

The model assumes that the following: (i) the top of the thick formation is closed, the thickness of the upper unutilized oil layer is h₁, the horizontal permeability is k_h1, the vertical permeability is k_v1, the integrated compression factor is C_t1, the porosity is φ₁, and the crude oil viscosity is μ_o; (ii) the bottom of the thick oil layer is closed, the thickness of the lower water layer sand body is h₂, i.e., the water-body multiple is n = h₂/h₁, the horizontal permeability is k_h2, the vertical permeability is k_v2, the integrated compression coefficient is C_t2, the porosity is φ₂, and the viscosity of the formation water is μ_w; (iii) The fluid is single-phase and slightly compressible. The gravity and capillary forces of the model are neglected. The percolation satisfies Darcy’s law; (iv) Horizontal well length is L_h, and the distance of the horizontal well from the oil–water contact is z_w. The model is produced at a production rate of q with an infinite inflow capacity.

2.2. Mathematical Modeling and Solving

The dimensionless seepage control equations can be represented as follows:

$$\{\begin{array}{l}\frac{{\partial }^2{p}_{\mathrm{D}1}}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2{p}_{\mathrm{D}1}}{\partial y_{\mathrm{D}}^2}+\frac{{\partial }^2{p}_{\mathrm{D}1}}{\partial z_{\mathrm{D}}^2}=\frac{\partial p_{\mathrm{D}1}}{\partial t_{\mathrm{D}}}\\ \frac{{\partial }^2{p}_{\mathrm{D}2}}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2{p}_{\mathrm{D}2}}{\partial y_{\mathrm{D}}^2}+\lambda \frac{{\partial }^2{p}_{\mathrm{D}2}}{\partial z_{\mathrm{D}}^2}=\frac{1}{\eta }\frac{\partial p_{\mathrm{D}2}}{\partial t_{\mathrm{D}}}\end{array}$$

(1)

The initial condition can be represented as follows:

$$p_{\mathrm{D}1}\left(x,y,z,t=0\right)=p_{\mathrm{D}2}\left(x,y,z,t=0\right)=0$$

(2)

The external boundary condition can be represented as follows:

$$p_{\mathrm{D}1}\left(r\to \infty ,z,t\right)=p_{\mathrm{D}2}\left(r\to \infty ,z,t\right)=0$$

(3)

The top-bottom closed boundary condition can be represented as follows:

$$\frac{\partial p_{\mathrm{D}1}\left(x,y,h_1-z_{\mathrm{w}},t\right)}{\partial z_{\mathrm{D}}}=\frac{\partial p_{\mathrm{D}2}\left(x,y,-h_2-z_{\mathrm{w}},t\right)}{\partial z_{\mathrm{D}}}=0$$

(4)

The coupling conditions at the oil–water contact can be represented as follows:

$$\{\begin{array}{l}{p}_{\mathrm{D}1}\left(x,y,-z_{\mathrm{w}},t\right)=p_{\mathrm{D}2}\left(x,y,-z_{\mathrm{w}},t\right)\\ \frac{K_{\mathrm{v}1}}{{\mu }_{\mathrm{o}}}\frac{\partial p_{\mathrm{D}1}\left(x,y,-z_{\mathrm{w}},t\right)}{\partial z_{\mathrm{D}}}=\frac{K_{\mathrm{v}2}}{{\mu }_{\mathrm{w}}}\frac{\partial p_{\mathrm{D}2}\left(x,y,-z_{\mathrm{w}},t\right)}{\partial z_{\mathrm{D}}}\end{array}$$

(5)

Define the dimensionless variable as $p_{\mathrm{D}1}=\frac{\sqrt{K_{\mathrm{h}1}{K}_{\mathrm{v}1}}{L}_{\mathrm{h}}\left(p_{\mathrm{i}}-p_1\right)}{1.842\times {10}^{-2}q{\mu }_{\mathrm{o}}}$; $p_{\mathrm{D}2}=\frac{\sqrt{K_{\mathrm{h}1}{K}_{\mathrm{v}1}}{L}_{\mathrm{h}}\left(p_{\mathrm{i}}-p_2\right)}{1.842\times {10}^{-2}q{\mu }_{\mathrm{w}}}$; $t_{\mathrm{D}}=\frac{3.6K_{\mathrm{h}1}t}{{\phi }_1{\mu }_{\mathrm{o}}{C}_{\mathrm{t}1}}$; $\eta =\left(\frac{K_{\mathrm{h}2}}{{\phi }_2{\mu }_{\mathrm{w}}{C}_{\mathrm{t}2}}\right)/\left(\frac{K_{\mathrm{h}1}}{{\phi }_1{\mu }_{\mathrm{o}}{C}_{\mathrm{t}1}}\right)$; $h_{\mathrm{D}1}=\frac{2h_1}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}1}}{K_{\mathrm{v}1}}}$; $h_{\mathrm{D}2}=\frac{2h_2}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}2}}{K_{\mathrm{v}2}}}$; $x_{\mathrm{D}}=\frac{2x}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}1}}{K_{\mathrm{v}1}}}$; $y_{\mathrm{D}}=\frac{2y}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}1}}{K_{\mathrm{v}1}}}$; $z_{\mathrm{D}}=\frac{2z}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}1}}{K_{\mathrm{v}1}}}$; $z_{\mathrm{wD}}=\frac{2z_{\mathrm{w}}}{L_{\mathrm{h}}}\sqrt{\frac{K_{\mathrm{h}1}}{K_{\mathrm{v}1}}}$; $L_{\mathrm{D}}=\frac{L_{\mathrm{h}}}{2h_1}\sqrt{\frac{K_{\mathrm{v}1}}{K_{\mathrm{h}1}}}$; $C_{\mathrm{D}}=\frac{2C}{\mathsf{\pi }{\phi }_1{C}_{\mathrm{t}1}{h}_1{L}_{\mathrm{h}}}$.

A Laplace transform on t_D for both ends of Equation (1) yields, and it is obtained that

$$\{\begin{array}{l}\frac{{\partial }^2{\overline{p}}_{\mathrm{D}1}}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2{\overline{p}}_{\mathrm{D}1}}{\partial y_{\mathrm{D}}^2}+\frac{{\partial }^2{\overline{p}}_{\mathrm{D}1}}{\partial z_{\mathrm{D}}^2}=s{\overline{p}}_{\mathrm{D}1}\\ \frac{{\partial }^2{\overline{p}}_{\mathrm{D}2}}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2{\overline{p}}_{\mathrm{D}2}}{\partial y_{\mathrm{D}}^2}+\lambda \frac{{\partial }^2{\overline{p}}_{\mathrm{D}2}}{\partial z_{\mathrm{D}}^2}=\frac{s}{\eta }{\overline{p}}_{\mathrm{D}2}\end{array}$$

(6)

Using the Green’s function method, Equation (6) is transformed into the following:

$$\{\begin{array}{l}\left(\frac{{\partial }^2}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2}{\partial y_{\mathrm{D}}^2}+\frac{{\partial }^2}{\partial z_{\mathrm{D}}^2}-s\right){\overline{G}}_1\left(s,r,r^{\prime }\right)=-\delta \left(r-r^{\prime }\right)\\ \left(\frac{{\partial }^2}{\partial x_{\mathrm{D}}^2}+\frac{{\partial }^2}{\partial y_{\mathrm{D}}^2}+\lambda \frac{{\partial }^2}{\partial z_{\mathrm{D}}^2}-\frac{s}{\eta }\right){\overline{G}}_2\left(s,r,r^{\prime }\right)=0\end{array}$$

(7)

A Fourier cosine transformation of x, y in Equation (7) yields, and it is obtained that

$$\{\begin{array}{l}\left(\frac{{\partial }^2}{\partial z_{\mathrm{D}}^2}-v_1^2\right){\widehat{\widehat{\overline{G}}}}_1=-\delta \left(r-r^{\prime }\right)\\ \left(\frac{{\partial }^2}{\partial z_{\mathrm{D}}^2}-v_2^2\right){\widehat{\widehat{\overline{G}}}}_2=0\end{array}$$

(8)

where,

$$v_i^2=\frac{1}{{\lambda }_i}\left({\alpha }^2+{\beta }^2+\frac{s}{{\kappa }_i}\right)$$

(9)

where $\alpha$ and β are the variables of x, and y in Fourier space, respectively.

The general solution of the above equations can be represented as follows:

$$\{\begin{array}{l}{\widehat{\widehat{\overline{G}}}}_1\left(s,\alpha ,\beta ,z\right)=\frac{\pi }{4v_1}{e}^{-v_1\left|z-z^{\prime }\right|}+A_1{\mathrm{e}}^{v_{1}z}+B_1{\mathrm{e}}^{-v_{1}z}\\ {\widehat{\widehat{\overline{G}}}}_2\left(s,\alpha ,\beta ,z\right)=A_2{\mathrm{e}}^{v_{2}z}+B_2{\mathrm{e}}^{-v_{2}z}\end{array}$$

(10)

Using the laws of transmission and reflection as pressure propagates through different reservoir media, according to Refs. [2,38], it is obtained that

$$\begin{array}{l}{\widehat{\widehat{\overline{G}}}}_1\left(s,\alpha ,\beta ,r_{\mathrm{w}}\right)=\frac{\pi }{4v_1}\frac{1}{1-{\mathsf{\Phi }}_{\mathrm{u}1}{\mathsf{\Phi }}_{\mathrm{d}1}{\mathrm{e}}^{-2v_1{h}_1}}\times [\left(1+{\mathsf{\Phi }}_{\mathrm{d}1}{\mathrm{e}}^{-2v_1{z}_w}\right){\mathsf{\Phi }}_{\mathrm{u}1}{\mathrm{e}}^{-2v_1\left(h_1-z_w\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\left(1+{\mathsf{\Phi }}_{\mathrm{u}1}{\mathrm{e}}^{-2v_1\left(h_1-z_w\right)}\right){\mathsf{\Phi }}_{\mathrm{d}1}{\mathrm{e}}^{-2v_1{z}_w}]\end{array}$$

(11)

where u and d are the upward and downward reflection coefficients, respectively.

Combining with the coupling conditions at the oil–water contact, it is obtained that

$${\mathsf{\Phi }}_{\mathrm{u}1}=1$$

(12)

$${\mathsf{\Phi }}_{\mathrm{d}1}={\mathrm{e}}^{-2v_2{h}_2}$$

(13)

Bring Equations (12) and (13) into Equation (11) and simplify to obtain the Green’s point source function within the oil layer in Fourier space.

$${\widehat{\widehat{\overline{G}}}}_1\left(s,\alpha ,\beta ,z,z^{\prime }\right)=\frac{\pi }{4v_1}{e}^{-v_1\left|z-z^{\prime }\right|}+\frac{\pi }{4v_1}\frac{{\left[{\mathrm{e}}^{-\left(v_{1}z+v_2{h}_2\right)}+{\mathrm{e}}^{-\left(v_1{h}_1-v_{1}z\right)}\right]}^2}{1-{\mathrm{e}}^{-2\left(v_1{h}_1+v_2{h}_2\right)}}$$

(14)

The pressure response solution for a horizontal well in Laplace space is obtained by taking two Fourier cosine inversions of the above equation and integrating them along the horizontal wellbore.

$${\overline{p}}_{\mathrm{wD}}\left(s,r_{\mathrm{wD}}\right)=\frac{2}{s}\left[{\overline{p}}_{\mathrm{wuD}}\left(s,r_{\mathrm{wD}}\right)+\frac{4}{{\pi }^2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}\alpha \frac{{\sin }^2\alpha }{{\alpha }^2}}{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{d}\beta {\widehat{\widehat{\overline{G}}}}_{\mathrm{h}1}\left(s,\alpha ,\beta ,r_{\mathrm{wD}}\right)}\right]$$

(15)

where,

$${\overline{p}}_{\mathrm{wuD}}\left(s,r_{\mathrm{wD}}\right)=\frac{1}{2}\left[\frac{K_0\left(r_{\mathrm{wD}}\sqrt{s}\right)}{r_{\mathrm{w}}\sqrt{s}{K}_0\left(r_{\mathrm{wD}}\sqrt{s}\right)}-\frac{1-{\mathrm{e}}^{-2\sqrt{s}}}{2\sqrt{s}}-E_1\left(2\sqrt{s}\right)\right]$$

(16)

$${\widehat{\widehat{\overline{G}}}}_{\mathrm{h}1}\left(s,\alpha ,\beta ,r_{\mathrm{wD}}\right)=\frac{\pi }{4v_1}\frac{{\left[{\mathrm{e}}^{-\left(v_1{z}_{\mathrm{wD}}+v_2{h}_{\mathrm{D}2}\right)}+{\mathrm{e}}^{-\left(v_1{h}_{\mathrm{D}1}-v_1{z}_{\mathrm{wD}}\right)}\right]}^2}{1-{\mathrm{e}}^{-2\left(v_1{h}_{\mathrm{D}1}+v_2{h}_{\mathrm{D}2}\right)}}$$

(17)

Considering the effect of wellbore storage and skin effect, according to Duhamel’s principle, it is obtained that

$${\overline{p}}_{\mathrm{wD}}=\frac{s{\overline{p}}_{\mathrm{wD}}+S}{s+s^2{C}_{\mathrm{D}}\left(s{\overline{p}}_{\mathrm{wD}}+S\right)}$$

(18)

Equation (18) is transformed to obtain the following:

$${\overline{q}}_{wD}=\frac{1}{s^2{\overline{p}}_{wD}}$$

(19)

By numerically inverting the above equation using the Stehfest method, a horizontal well pressure response curve for a bottom-water reservoir with finite water bodies can be plotted, and the transient productivity of a horizontal well can also be calculated [39].

3. Typical Curve Analysis

3.1. Model Verification

As shown in Figure 2, the type curves of pressure obtained by the Saphir numerical solution and the analytical solution proposed in this paper are anastomotic when water-body multiples are set to 50. The curves show the common characteristics. The pressure derivative curve is a 0.5 horizontal line at the early vertical radial flow stage, the slope of the pressure derivative curve is −0.5 at the hemispherical flow stage, while it is an L_D/51 horizontal line at the late system pseudo-radial flow stage. Thus, the model and solution in the paper are reliable.

3.2. Typical Curve

The newly established model can solve the pressure in constant production. Therefore, the parameters were selected as the dimensionless wellbore storage factor C_D = 0.00001, the skin factor S = 0.1, the horizontal well dimensionless length L_D = 3.2, and the dimensionless distance between the horizontal well and the oil–water contact z_wD = 0.5. The pressure and pressure derivative of a horizontal well in bottom-water reservoirs with different water-body multiples (n = 5, 20, and 100) are calculated by using the model. The typical curves of pressure and pressure derivative for horizontal wells in bottom-water reservoirs with limited water-body multiples are drawn (Figure 3). The results calculated for the conventional top-bottom closed reservoir model and the infinite rigid bottom-water reservoir model (top closed, constant pressure at the oil–water contact) are plotted on the same plate for comparison and analysis of their flow regimes (Figure 3).

From the three sets of double logarithmic curves for n = 5, 20, and 100 in Figure 3, it can be seen that there are six main flow regimes in the seepage process of horizontal wells in bottom water reservoirs with finite water bodies. (I) Early wellbore storage effect stage: the pressure and pressure derivative curves in this stage show a straight line with a slope of 1; (II) transition section between the wellbore storage effect stage and the early vertical radial flow stages: the larger the skin factor, the higher the hump; (III) early vertical radial flow stage: fluid flows in the vertical plane towards the wellbore (Figure 4a), and the pressure derivative curve in this stage shows a horizontal line with a value of 0.5; (IV) transition section between the early vertical radial flow phase and the hemispherical flow stage: pressure waves gradually propagate from the vicinity of the horizontal wellbore towards the top closed boundary; (V) hemispherical flow stage: the pressure wave propagates continuously in the direction of the bottom, and the fluid flows in a hemispherical flow in the vertical direction towards the horizontal well before propagating to the bottom closed boundary (Figure 4b). The hemispherical flow stage lasts longer as the water-body multiple increases, and the pressure derivative curve gradually shows a straight line with a slope of −0.5; (VI) late system pseudo-radial flow stage: after the pressure wave propagates to the bottom closed boundary, the pressure propagates in the plane in an (elliptical) circle to the distance, and the reservoir fluid flows in the form of pseudo-radial flow to the horizontal well (Figure 4c). The pressure derivative curve in this stage presents a horizontal line with the value of L_D/(n + 1), which is a unique characteristic of the horizontal well of a bottom-water reservoir with finite water bodies. The size of the water-body multiple in the bottom-water reservoirs with finite water bodies can be determined by the ratio of the pressure derivative value of late system pseudo-radial flow to the pressure derivative value of early vertical radial flow, whose value is 2L_D/(n + 1).

As can be seen from the double logarithmic curves of the top-bottom closed common reservoir in Figure 3, there are five main flow regimes in the horizontal well seepage process in top-bottom closed normal reservoirs. The first three stages are consistent with a finite water body bottom-water reservoir, and the last two stages are different. This is consistent with the existing research conclusions, which proves the rationality of the results in this paper [39]. For conventional top-bottom closed reservoirs, the fourth stage (IV) is the linear flow stage. This stage begins after the fluid flows in the vertical plane towards the wellbore and reaches the top-bottom boundary, which is a linear flow of fluid between the upper and lower reservoir boundaries, with the pressure derivative curve showing a straight-line segment with a slope of 0.5. The fifth stage (V) is the late pseudo-radial flow stage: the pressure wave has propagated farther away from the wellbore and approximates a straight well in production relative to the entire seepage range, with fluids converging on the wellbore from all directions to form a radial flow, and the pressure derivative curve is horizontal at this stage.

As can be seen from the double logarithmic curve of the top-closed, oil–water contact constant pressure reservoir in Figure 3, there are five main flow regimes in the seepage process of a horizontal well in an infinite rigid bottom-water reservoir. The first four stages are consistent with a finite water body bottom-water reservoir; the fifth stage is different. For top-closed, constant-pressure reservoirs at the oil–water contact, the fifth stage (V) is the steady flow stage. After the pressure wave propagates to the bottom constant pressure boundary, the flow pressure at the bottom of the horizontal well gradually stabilizes, the pressure at various points in the reservoir also stabilizes, the pressure curve tends to a horizontal line, and the pressure derivative curve falls rapidly to zero.

Comparing the three types of double logarithmic curves in Figure 3 and the above analysis, it can be seen that, unlike the conventional top-bottom closed reservoir and the infinite rigid bottom-water reservoir horizontal well seepage process, the influence of finite water bodies on the flow characteristics of horizontal wells mainly occurs during and after the propagation of pressure waves to the bottom boundary, where the double logarithmic curves appear significantly different. As the number of water-body multiple increases, the pressure curve and pressure derivative curve of horizontal wells in the bottom-water reservoir with finite water bodies increasingly deviate from the pressure curve and pressure derivative curve of horizontal wells in conventional top-bottom closed reservoirs and increasingly converge to the pressure curve and pressure derivative curve of horizontal wells in infinitely large rigid water bodies. The pressure and pressure derivatives of horizontal wells in bottom-water reservoirs with finite water bodies are between those calculated by the conventional top-bottom closed reservoir model and the infinite rigid bottom-water reservoir model, in line with objective laws.

4. Analysis of Factors Influencing Productivity

4.1. Water-Body Multiplier

The parameters were selected as the dimensionless wellbore storage factor C_D = 0.00001, the skin factor S = 0.1, the horizontal well dimensionless length L_D = 3.2, and the dimensionless distance between the horizontal well and the oil–water contact z_wD = 0.5. The effect of different water-body multiples (n = 0, 5, 10, 20, and 50) on the production of horizontal wells was calculated using the model, and the results are shown in Figure 5. It can be seen that, (i) at the early stage of horizontal well production, horizontal well dimensionless production decreases with the increase in dimensionless time. The production curves of different water-body multiples overlap, showing no effect of water-body multiples on production. The duration of this stage is relatively short. At this stage, the pressure wave does not propagate to the oil–water contact and is not influenced by the energy of the water body; the process is a depletion development, and the production of the horizontal well declines. (ii) As the time of dimensionless production increases, the dimensionless production of horizontal wells gradually decreases. The overall production curve of horizontal well declines due to a lack of energy for finite water bodies, as opposed to infinite rigid bottom-water reservoirs. (iii) The higher the finite water-body multiple, the smaller the decline in yield. As the water-body multiple increases, the capacity of the water body to replenish energy increases, and the decline in yield decreases and gradually levels off.

4.2. Distance between Horizontal Well and Oil–Water Contact

The parameters were set as the dimensionless wellbore storage factor C_D = 0.00001, the skin factor S = 0.1, the horizontal well dimensionless length L_D = 3.2, and the finite water-body multiple n = 50. The model was used to calculate the effect of different dimensionless distances (z_wD =0.3, 0.5, 0.7, and 0.9) between the horizontal well and the oil–water contact on the production of the horizontal well. The results of which are shown in Figure 6. It can be seen that, (i) at the beginning of horizontal well production, horizontal well dimensionless production decreases with the increase in the dimensionless time. The production curves at different distances from the horizontal well and oil–water contact overlap. At this stage, the pressure wave does not propagate to the oil–water contact, and the horizontal well is a depletion development with declining production. (ii) As the time of dimensionless production increases, the dimensionless production of horizontal wells gradually decreases. Unlike infinite rigid bottom-water reservoirs, the overall performance is characterized by a lack of energy in finite water bodies and a decline in horizontal well production. (iii) The smaller the distance between the horizontal well and the oil–water contact, the smaller the rate of production declines. The closer the horizontal well is to the oil–water contact, the greater the pressure gradient acting on the horizontal well, the greater the energy supply, and the smaller the production decline.

4.3. Skin Factor

The parameters selected were the dimensionless wellbore storage factor C_D = 0.00001, the dimensionless distance between the horizontal well and the oil–water contact z_wD = 0.5, the horizontal well dimensionless length L_D = 3.2, and the finite water-body multiple n = 50. The model was used to calculate the effect of the skin factor (S = 0.5, 2.5, and 5) on the production of horizontal wells, and the results are shown in Figure 7. It can be seen that, (i) the dimensionless production of horizontal wells gradually decreases as the dimensionless production time increases. Unlike infinite rigid bottom-water reservoirs, the overall performance is characterized by a lack of energy in finite water bodies and a decline in horizontal well production. (ii) The larger the skin factor, the lower the initial production of the horizontal well. The larger the skin factor, the greater the seepage resistance in the near-well zone, the greater the additional pressure drop, the smaller the pressure gradient acting on the horizontal well, and the lower the horizontal well production. (iii) The skin factor affects the whole process of transient productivity from horizontal wells in bottom-water reservoirs with limited water bodies, but the skin effect has a large impact on production in the early stages of production and becomes less influential in the later stages. This is because the mechanical skin mainly acts in the near-well zone, affecting the seepage capacity of the near-well area. The pressure relief of the near-well zone mainly appears at the early stage of horizontal well production, so it has a more significant impact on early production; with the extension of production time, the impact of the skin on production gradually decreases after the pressure wave propagates to the far-well zone.

4.4. Field Application

Well A1 is a horizontal well in the LD oilfield in Bohai Bay. The water-body multiple is 18, the total length of the horizontal section of the horizontal well is 456 m, the well radius is 0.12 m, the formation thickness is 8 m, the horizontal well is 6 m from the oil–water contact, the porosity is 32%, the permeability is 1500 mD, the comprehensive compression coefficient is 0.00098, the oil volume coefficient is 1.08, and the formation oil viscosity is 8 mPa·s. The model and method proposed in this paper are used for prediction, resulting in a good agreement between the calculated data and the actual data, as seen in Figure 8.

Overall, the model and method proposed in this paper can be used to effectively perform the production analysis of a horizontal well in a bottom-water reservoir with finite water bodies, which broadens the research means of productivity and is more convenient and faster than numerical simulation in the analysis of different parameters.

5. Conclusions

The commonly used horizontal well production model for bottom-water reservoirs treats bottom-water as an infinite rigid body of water and does not consider the effect of the size of the water body. In response to the above problem, a semi-analytical model of the coupled flow between the oil formation and the finite water body was developed using Green’s function and potential superposition method. Moreover, the effects of factors including water-body multiples, horizontal well locations, and skin effect on the production of horizontal wells in bottom-water reservoirs with finite water bodies were studied.

Seepage in horizontal wells in bottom-water reservoirs with finite water bodies can be divided into six main flow regimes. As the water-body multiple increases, the hemispherical flow duration is longer, the pseudo-radial flow in the later stage system appears later, and the pressure derivative value is lower. This law can be used for the pressure dynamic analysis of horizontal wells in this type of reservoir.

Production from horizontal wells in bottom-water reservoirs with finite water bodies is influenced by the water-body multiple, horizontal well location, and skin factor. The larger the finite water-body multiple and the smaller the distance between the horizontal well and the oil–water contact, the smaller the decline in production. The larger the skin factor, the lower the initial production of the horizontal well, the greater the early-stage impact, and the smaller the later-stage impact. For reservoirs with a defined water-body multiple, optimizing the distance between the horizontal well and the oil–water contact and improving the drilling process to reduce the surface skin are important ways to increase the production of horizontal wells in bottom-water reservoirs with limited water bodies.